A Comparison of Atomistic and Surface Enhanced Continuum Approaches at Finite Temperature

  • Denis Davydov
  • Ali Javili
  • Paul Steinmann
  • Andrew McBride
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 30)


The surface of a continuum body generally exhibits properties that differ from those of the bulk. Surface effects can play a significant role for nanomaterials, in particular, due to their large value of surface-to-volume ratio. The effect of solid surfaces at the nanoscale is generally investigated using either atomistic or enhanced continuum models based on surface elasticity theory. Hereby the surface is equipped with its own constitutive structure. Atomistic simulations provide detailed information on the response of the material. Discrete and continuum systems are linked using averaging procedures which allow continuum quantities such as stress to be obtained from atomistic calculations. The objective of this contribution is to compare the numerical approximations of the surface elasticity theory to a molecular dynamics based atomistic model at finite temperature. The bulk thermo-elastic parameters for the continuum’s constitutive model are obtained from the atomistic simulation. The continuum model takes as its basis the fully nonlinear thermo-elasticity theory and is implemented using the finite element method. A representative numerical simulation of face-centered cubic copper confirms the ability of a surface enhanced continuum formulation to reproduce the behaviour exhibited by the atomistic model, but at a far reduced computational cost.


Representative Volume Element Deformation Gradient Atomistic Simulation Reference Configuration Continuum Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The first author is grateful to the German Science Foundation (Deutsche Forschungs-Gemeinschaft), grant STE 544/46-1, for their financial support. The support of this work by the ERC Advanced Grant MOCOPOLY is gratefully acknowledged by the second and third authors.


  1. 1.
    Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecular systems. J. Elast 100, 63–143 (2010)CrossRefGoogle Scholar
  2. 2.
    Daher, N., Maugin, G.A.: The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces. Acta Mech. 60(3–4), 217–240 (1986)CrossRefGoogle Scholar
  3. 3.
    Davydov, D., Javili, A., Steinmann, P.: On molecular statics and surface-enhanced continuum modeling of nano-structures. Comput Mater Sci [accepted]Google Scholar
  4. 4.
    Davydov, D., Steinmann, P.: Reviewing the roots of continuum formulations in molecular systems. Part I: Particle dynamics, statistical physics, mass and linear momentum balance equations. Math Mech Solids [accepted]Google Scholar
  5. 5.
    Foiles, S.M., Baskes, M.I., Daw, M.S.: Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B 33(12), 7983–7991 (1986)CrossRefGoogle Scholar
  6. 6.
    Gibbs, J.W.: The Scientific Papers of JW Gibbs, vol. 1. Dover Publications, New York (1961)Google Scholar
  7. 7.
    Green, M.S.: Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids. J. Chem. Phys. 22(3), 398–413 (1954)CrossRefGoogle Scholar
  8. 8.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)CrossRefGoogle Scholar
  9. 9.
    Hardy, R.: Formulas for determining local properties in molecular-dynamics simulations—shock waves. J. Chem. Phys. 76, 622–628 (1982)CrossRefGoogle Scholar
  10. 10.
    Irving, J., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)CrossRefGoogle Scholar
  11. 11.
    Javili, A., McBride, A., Mergheim, J., Steinmann, P., Schmidt, U.: Micro-to-macro transitions for continua with surface structure at the microscale. SubmittedGoogle Scholar
  12. 12.
    Javili, A., Steinmann, P.: On thermomechanical solids with boundary structures. Int. J. Solids Struct. 47(24), 3245–3253 (2010)CrossRefGoogle Scholar
  13. 13.
    Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part III: The thermomechanical case. Comput. Methods Appl. Mech. Eng. 200(21–22), 1963–1977 (2011)CrossRefGoogle Scholar
  14. 14.
    Jones, W., March, N.H.: Theoretical Solid State Physics. Dover Publications, New York (1985)Google Scholar
  15. 15.
    Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12(6), 570–586 (1957)CrossRefGoogle Scholar
  16. 16.
    Noll, W.: Die herleitung der grundgleichungen der thermomechanick der kontinua aus der statistichen mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)Google Scholar
  17. 17.
    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995). Google Scholar
  18. 18.
    Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B Condens. Matter Mater. Phys. 71(9), 094104 (2005)Google Scholar
  19. 19.
    Zimmerman, J.A., Jones, R.E., Templeton, J.A.: A material frame approach for evaluating continuum variables in atomistic simulations. J. Comput. Phys. 229, 2364–2389 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Denis Davydov
    • 1
  • Ali Javili
    • 1
  • Paul Steinmann
    • 1
  • Andrew McBride
    • 2
  1. 1.LTMUniversity of Erlangen–NurembergErlangenGermany
  2. 2.CERECAMUniversity of Cape TownRondeboschSouth Africa

Personalised recommendations